Method for estimating immeasurable process variables during a series of discrete process cycles

ABSTRACT

A method for estimating a process variable associated with a series of operations of a manufacturing process includes deriving a model that represents a given operation of the manufacturing process. The operation has first, second, and third process variables associated therewith. The model includes the first, second, and third process variables. Variations in the first and second process variables during each of the operations are substantially immeasurable. The method further includes measuring the first process variable after a first one of the operations and measuring the third process variable during a second one of the operations using a sensing device. The method further includes estimating at least one of the first and second process variables during the second one of the operations using the measured first process variable, the measured third process variable, and the model. Additionally, the method includes controlling the second operation based on the at least one of the first and second estimated process variables.

CROSS-REFERENCE TO RELATED APPLICATION

This application claims the benefit of U.S. Provisional Application No.61/111,817 filed on Nov. 6, 2008. The disclosure of the aboveapplication is incorporated herein by reference in its entirety.

FIELD

The present disclosure relates to systems and methods for estimating aprocess variable associated with a series of operations of amanufacturing process.

BACKGROUND

Most industry processes for producing multiple parts of identical designand specifications involve repeating similar or identical process cyclesin series. Batch processing for biochemical, semiconductor, andmaterials industries as well as traditional manufacturing operationsincluding machining processes belong to this category. From a series ofprocess cycles, two streams of data can be obtained. For instance, inthe machining process, various sensors are used for in-processmeasurement of process variables such as powers, forces, and vibration.In contrast, the qualities of machined parts such as the surface finishand the tool conditions can be measured only by the post-processinspection in most applications. Despite recent progress, the real-timemeasurement of the condition of a grinding wheel is still a verychallenging task. Active research is taking place for the development ofsensors for in-process measurement of part qualities such as theresidual stress and surface finish. However, their industry-wideacceptance has not yet been realized.

Conventional approaches to monitoring and controlling a series ofprocess cycles tend to rely on only one out of the two data streams forthis purpose. Existing estimation schemes for estimating part qualitiesand tool conditions in real-time for the machining process have focusedonly on analyzing sensor signals while overlooking the significance ofpost-process data flow in batch production. The systems and methods ofthe present disclosure improve observability by supplementing in-processsensor signals with post-process measurement of the part quality andtool condition from previous cycles. Accordingly, the systems andmethods of the present disclosure utilize the post-process measurementdata to improve the estimation performance of process cycles.

This section provides background information related to the presentdisclosure which is not necessarily prior art.

SUMMARY

A method for estimating a process variable associated with a series ofoperations of a manufacturing process comprises deriving a model thatrepresents a given operation of the manufacturing process. The operationhas first, second, and third process variables associated therewith. Themodel includes the first, second, and third process variables.Variations in the first and second process variables during each of theoperations are substantially immeasurable. The method further comprisesmeasuring the first process variable after a first one of the operationsand measuring the third process variable during a second one of theoperations using a sensing device. The method further comprisesestimating at least one of the first and second process variables duringthe second one of the operations using the measured first processvariable, the measured third process variable, and the model.Additionally, the method comprises controlling the second operationbased on the at least one of the first and second estimated processvariables.

BRIEF DESCRIPTION OF DRAWINGS

The present disclosure will become more fully understood from thedetailed description and the accompanying drawings.

FIG. 1 illustrates a batch production of N parts that are processed inseries on a grinding machine.

FIG. 2 is a functional block diagram of a manufacturing system accordingto the present disclosure.

FIG. 3 is a functional block diagram of a machine control moduleaccording to the present disclosure.

FIG. 4 illustrates a method for estimating a process variable associatedwith a series of operations of a manufacturing process according to thepresent disclosure.

FIG. 5 is a schematic of a cylindrical grinding process.

FIG. 6 illustrates a comparison of estimation results based on twomeasurement settings with which the observability was tested.

FIG. 7 illustrates performance of the proposed scheme in estimating R₀as well as in predicting the surface roughness at the end of eachgrinding cycle.

FIG. 8 illustrates the overall schematics of the estimation algorithm.

FIG. 9 presents the results of estimating state variables V_(w)′ and valong with model parameter s₀ based on two measurement settings withexperimental data from the first batch.

FIG. 10 shows the results of estimation and prediction for the surfaceroughness with experimental data from the first batch.

FIG. 11 shows results of estimation for the wheel diameter withexperimental data from the first batch.

FIG. 12 shows estimation of V_(w)′, s₀, and v based on two measurementsettings with experimental data from the second batch.

FIG. 13 shows results of estimation and prediction for the surfaceroughness with experimental data from the second batch.

FIG. 14 shows results of estimation for the wheel diameter withexperimental data from the second batch.

FIG. 15 shows estimation of V_(W)′, s₀, and v with experimental datafrom a mix of different grinding cycles.

FIG. 16 shows results of estimation and prediction for the surfaceroughness with experimental data from a mix of different grindingcycles.

FIG. 17 shows grinding power P versus parameter τv_(s)v obtained fromstep responses of P.

FIG. 18 shows Parameter P/(K_(s)d_(w)v) versus accumulated metal removalV_(w)′ from step responses of P.

FIG. 19 shows surface roughness versus equivalent chip thicknessimmediately after wheel dressing.

FIG. 20 shows wheel profiles for different values of equivalent chipthickness.

FIG. 21 shows change of G-ratio with varying equivalent chip thickness.

DETAILED DESCRIPTION

The following description is merely exemplary in nature and is in no wayintended to limit the disclosure, its application, or uses. For purposesof clarity, the same reference numbers will be used in the drawings toidentify similar elements. As used herein, the phrase at least one of A,B, and C should be construed to mean a logical (A or B or C), using anon-exclusive logical OR. It should be understood that steps within amethod may be executed in different order without altering theprinciples of the present disclosure.

Batch production is commonly employed in industry to manufacture a groupof parts or products with identical design and specifications. The startof a new batch is marked by the launch of a new design, tool change, orarrival of a new lot from suppliers or preceding processes. From acontrol point of view, the start of a new batch normally coincides withsignificant changes in the process dynamics, and hence the states andmodel parameters are updated when a new batch starts.

FIG. 1 shows a schematic of batch production when N parts are processedin series on a grinding machine. Many machine tools in modern industryare equipped with various sensors for monitoring process variables suchas the grinding power, which are generally sampled at a constantfrequency. In contrast, the quality of each part is usually onlymeasured at a post-process inspection after its grinding cycle iscompleted. Ignoring the idle time between two consecutive grindingcycles, a series of grinding operations can be viewed as a continuousprocess with two output streams sampled at two distinct intervals. Inthe present disclosure, the sampling time of the sensor signal (T_(f))is set constant, and that of the part quality is T_(i)(>T_(f)), whichcorresponds to the cycle time of the ith grinding cycle, as shown inFIG. 1. While the following description is provided with reference to agrinding operation, it is readily understood that the estimationtechniques are applicable to other machining processes as well as othertypes of discrete processes.

Referring now to FIG. 2, a manufacturing system 100 includes a machinetool 102 and a machine control module 104. The machine control module104 actuates the machine tool 102 to perform an operation on parts 106-1and 106-2 (collectively “parts 106”) during a manufacturing process. Themachine tool 102 may include a grinding wheel 108. Accordingly, themachine tool 102 may be a grinding machine tool (e.g., a plungegrinder). While the machine tool 102 is described as a grinding machinetool that performs a grinding operation, the systems and methods of thepresent disclosure may be applicable to other machine tools that performother operations. For example, the systems and methods may be applicableto a milling machine that performs milling operations, a drillingmachine that performs drilling operations, and/or a lathe that performsa lathing operation. Additionally, the systems and methods of thepresent disclosure may be applicable to other tools and equipment thatperform other process that do not include machining operations. Forexample, the systems and methods of the present disclosure may beapplicable to batch processing for biochemical, semiconductor, andmaterials processes.

The machine control module 104 controls one or more actuators 110-1, . .. , and 110-n (collectively “actuators 110”) of the machine tool 102 toperform various operations on the parts 106. For example, the machinecontrol module 104 may control the actuators 110 to control a rotationalspeed of the grinding wheel 108, an infeed rate of the grinding wheel108, etc.

As used herein, the term module may refer to, be part of, or include anApplication Specific Integrated Circuit (ASIC), an electronic circuit, aprocessor (shared, dedicated, or group) and/or memory (shared,dedicated, or group) that execute one or more software or firmwareprograms, a combinational logic circuit, and/or other suitablecomponents that provide the described functionality.

One or more sensors 112-1, . . . , and 112-n (collectively “sensors112”) of the machine tool 102 measure process variables associated withthe manufacturing process. More specifically, the sensors 112 measureprocess variables associated with the machine tool 102 while the machinetool 102 is performing an operation. Process variables that may bemeasured by the sensors 112 during operation of the machine tool arereferred to hereinafter as “measurable process variables.” For example,measurable process variables associated with the machine tool 102 mayinclude the grinding power, a reduction in the size of the part 106,etc. The machine control module 104 may control the machine tool 102based on feedback signals received from the sensors 112. Accordingly,the machine control module 104 may control the rotational speed of thegrinding wheel 108 and the infeed rate of the grinding wheel 108 basedon the grinding power and the reduction in the size of the part 106.

Other processing variables associated with the manufacturing process maybe substantially immeasurable during the operation of the machine tool102. For example, measurements associated with the grinding wheel 108and/or the part 106 during the grinding process may be substantiallyimmeasurable. Process variables that are substantially immeasurableduring operation of the machine tool 102 are referred to hereinafter as“immeasurable process variables.” For example, immeasurable processvariables may include a condition of the grinding wheel 108 (e.g., adiameter of the grinding wheel 108), a residual stress associated withthe part 106, a roundness of the part 106, and a surface finish of thepart 106.

Immeasurable process variables may be measured after operation of themachine tool 102. The machine control module 104 may control the machinetool 102 during subsequent operations based on immeasurable processvariables which were measured after previous operations. Accordingly,the machine control module 104 may control a grinding operation based onprocess variables measured during the grinding operation using feedbackfrom the sensors 112 and immeasurable process variables which weremeasured prior to the current operation.

A machine tool measuring device 114 measures immeasurable processvariables associated with the machine tool 102 after a machiningoperation. In other words, after a grinding operation is complete, themachine tool measuring device 114 may measure process variablesassociated with the machine tool 102 that were substantiallyimmeasurable during the previous grinding operation. For example, themachine tool measuring device 114 may measure the condition of thegrinding wheel 108 (e.g., the diameter of the grinding wheel 108). Themeasurements taken by the machine tool measuring device 114 are fed backto the machine control module 104. Accordingly, the machine controlmodule 104 may actuate the machine tool 102 during subsequent operationsbased on measurements of the machine tool 102 taken after previousoperations.

A part measuring device 116 measures the immeasurable process variablesassociated with parts 106 machined by the machine tool 102. In otherwords, after a grinding operation is complete, the part measuring device116 may measure process variables associated with the part 106 that themachine tool 102 produced that were substantially immeasurable duringthe grinding operation. For example, the part measuring device 116 maymeasure the residual stress associated with the part 106, the roundnessof the part 106, and the surface finish of the part 106. Themeasurements taken by the part measuring device 116 are fed back to themachine control module 104. Accordingly, the machine control module 104may actuate the machine tool 102 during subsequent operations based onmeasurements of the parts 106 taken after previous operations. In FIG.2, a part 106-2 produced by a previous operation (operation N) ismeasured by the part measuring device 116. The measurements associatedwith the part 106-2 produced by operation N are fed back to the machinecontrol module 104. The machine control module 104 then controls themachine tool 102 to perform a subsequent operation (operation N+1) on asubsequent part 106-1 based on the measurements associated with the part106-2 produced by operation N.

The manufacturing system 100 may include a human-machine interface (HMI)120 that receives user input from a human user of the machine tool 102.The machine control module 104 may control the machine tool 102 based onthe user input. The HMI 120 may also display information associated withoperation of the machine tool 102 to the user. In some implementations,the user of the machine tool 102 may measure the immeasurable processvariables associated with the machine tool 102 and/or the parts 106produced and input the measurements into the HMI 120. Accordingly, themachine control module 104 may control the machine tool 102 based onimmeasurable process variables measured by the user.

Referring now to FIG. 3, the machine control module 104 includes aninput module 122, a post-process acquisition module 124, an in-processacquisition module 126, an estimation module 128, and an actuationmodule 130. The input module 122 receives user input from the HMI 120.The post-process acquisition module 124 receives data from the partmeasuring device 116 corresponding to measurements of parts 106 takenbetween operations of the machine tool 102. The post-process acquisitionmodule 124 also receives data from the machine tool measuring device 114corresponding to measurements of the machine tool 102 taken betweenoperations of the machine tool 102. The post-process acquisition module124 determines the immeasurable process variables based on the datareceived from the machine tool measuring device 114 and/or the partmeasuring device 116. For example, the post-process acquisition module124 may determine the condition of the grinding wheel 108 (e.g., thediameter of the grinding wheel 108), the residual stress associated withthe part 106, the roundness of the part 106, and the surface finish ofthe part 106 based on the data received from the machine tool measuringdevice 114 and/or the part measuring device 116.

The in-process acquisition module 126 receives signals from sensors 112that measure process variables during operation of the machine tool 102.The in-process acquisition module 126 determines the measurable processvariables based on the data received from the sensors 112 duringoperations of the machine tool 102. For example, the in-processacquisition module 126 may determine the grinding power and thereduction in the size of the part 106 based on data received from thesensors 112.

The estimation module 128 includes a model of the operations associatedwith the machine tool 102. For example, the estimation module 128 mayinclude a state-space model representation of the operations (e.g.,grinding operations) associated with the machine tool 102. The modelincludes the measurable and immeasurable process variables. The model isdescribed hereinafter in further detail. The estimation module 128 mayimplement an estimation scheme (e.g., an estimation algorithm) todetermine the immeasurable process variables during operations of themachine tool 102. For example, the estimation module 128 may implementan estimation scheme based on extended Kalman filters. The estimationscheme is described hereinafter in further detail.

In some implementations, the estimation module 128 includes a model thatmodels variations in parameters of the machine tool 102 and/or the parts106 between operations. For example, the model may include one or morenoise terms that model the variations. The model that includes the noiseterms is described hereinafter in further detail. A filter algorithm(e.g., a Kalman filter) may be derived based on the model that modelsvariations between operations.

Variations in parameters of the machine tool 102 may include a variationin the radius of the grinding wheel 108 between the end of a prioroperation and the beginning of a subsequent operation (e.g., due totemperature changes). Other variations in parameters of the machine tool102 may also include, for example, variations that affect the positionof the part 106 held within the machine tool 102. The position of thepart 106 may vary between operations due to tolerances of a chuck thatholds the part 106 in the machine tool 102. A magnitude of the noiseterm may be based on an amount of expected variation of a parameter ofthe machine tool 102. For example, the magnitude of the noise termcorresponding to a parameter of the machine tool 102 may be greater whenthe tolerances related to the parameter are wider.

Variations in parameters associated with the parts 106 betweenoperations may be due to variations in material properties of the parts106 between operations. Material properties that vary between operationsmay include a hardness of the part 106, strength of the part 106, aductility of the part 106, and a location and amount of imperfections inthe part 106. Additionally, variations in conditions of the parts 106between operations may be due to a difference in initial sizes of theparts 106.

The model may also account for variations in model parameters thatrepresent physical properties of the manufacturing system 100. Forexample, model parameters related to mass of components of the machinetool 102 and/or the parts 106 may be modified by a noise term in orderto represent variations of the physical properties of the manufacturingsystem 100.

The estimation module 128 may estimate the immeasurable processvariables during an operation of the machine tool 102 based on themodel, the measurable process variables measured during the operation,and the immeasurable process variables measured after a prior operationof the machine tool 102. The actuation module 130 controls the operationof the machine tool 102 during the operation based on the estimatedimmeasurable process variables. Accordingly, the actuation module 130actuates the machine tool 102 during the operation based on theimmeasurable process variables that were previously measured.

Manufacture of a first and second part in the manufacturing system 100is now described. The machine control module 104 actuates the machinetool 102 to produce the first part during a first operation. The partmeasuring device 116 and/or the machine tool measuring device 114measure the first part and or components of the machine tool 102,respectively, after the first operation to determine an immeasurableprocess variable. Additionally or alternatively, the user may input theimmeasurable process variable based on measurements of the first partand or the machine tool 102 after the first operation. In someimplementations, the machine control module 104 may modify gainparameters of the estimation algorithm using a noise term before a startof a second operation.

The second part is then loaded into the machine tool 102. The machinecontrol module 104 actuates the machine tool 102 to perform the secondoperation on the second part. The sensors 112 of the machine tool 102feed back data to the machine control module 104. The machine controlmodule 104 may control the second operation based on the data fed backfrom the sensors 112 during the second operation. Additionally, themachine control module 104 controls the second operation based on theimmeasurable process variables that were measured after the firstoperation. The machine control module 104 estimates the immeasurableprocess variables during the second operation using the measured processvariables (i.e., data fed back from the sensors 112) during the secondoperation, the immeasurable process variables measured after the firstoperation, and the model. The machine control module 104 controls themachine tool 102 during the second operation based on the estimatedimmeasurable process variables.

Referring now to FIG. 4, a method for estimating a process variableassociated with a series of operations of a manufacturing process startsat 200. At 200, a model is derived that represents a series ofoperations of a manufacturing process. At 202, the machine tool 102performs a first operation on a first part. At 204, the machine toolmeasuring device 114 and/or the part measuring device 116 measure aprocess variable (V₁) that was substantially immeasurable during thefirst operation. At 205, the estimation module 128 may modify parametersof the estimation algorithm using noise terms. At 206, the machine tool102 starts a second operation on a second part. At 208, the in-processacquisition module 126 determines a process variable (V₂) during thesecond operation based on feedback from sensors 112. At 210, theestimation module 128 estimates the value of V₁ during the secondoperation based on the measured V₁ at 204, the measured V₂ at 208, andthe model. At 212, the machine control module 104 controls the machinetool 102 during the second operation based on the estimated value of V₁.

A derivation of a state-space model from existing analytical models ofthe cylindrical plunge grinding process is briefly described herein.FIG. 5 shows a schematic of a cylindrical grinding process, in which arotating cylindrical work-piece with a nominal diameter of d_(w) and asurface velocity of v_(w) is ground by a rotating grinding wheel with anominal diameter of d_(s) and a surface velocity of v_(s). The grindingwheel is fed into the work-piece at a command infeed rate, u.

Three dynamic relationships may be included for the cylindrical grindingprocess in an analytical model. It may be assumed that the grinding iscarried out in a chatter-free region. The first relationship is thedynamic delay of the actual infeed rate, v (mm/s), in response to thecommand infeed rate, u (mm/s), due to the mechanical stiffness andsharpness of the wheel surface, which is frequently modeled as afirst-order system:

{dot over (v)}=(u−v)/τ  (1)

where τ(s) is the time constant whose value is dependent on themachine-wheel-workpiece stiffness and the sharpness of the wheel. Thesharpness of the wheel decreases with the accumulated amount of materialremoved after a tool change, V_(w)′(mm³/mm), due to attrition of thegrits. The accumulated metal removal, by its definition, is related toinfeed rate v by another first-order differential equation:

{dot over (V)}_(w)′=πd_(w)v  (2)

On the other hand, the radial wheel wear—which involves a progressivereduction in the diameter of the grinding wheel—may be obtained bymanipulation of an analytical model represented by the followingequation:

$\begin{matrix}{{\overset{.}{d}}_{s} = {{- \frac{2\pi^{g}d_{w}^{1 + g}}{d_{s_{0}}G_{1}}}v_{s}^{- g}v^{1 + g}}} & (3)\end{matrix}$

where d_(s) ₀ is the initial wheel diameter (mm), and G₁ and g are modelparameters.

Based on (1)-(3), three state variables are defined to describe thedynamic relationships in the grinding process using the following stateequation:

{dot over (x)}=f(x,u)+η(t)  (4)

where x=(x₁, x₂, x₃)^(T)=(V_(w)′, v, d_(s))^(T)ε

³u=(u₁, u₂)^(T)=(u, v_(s))^(T)ε

²and η(t) ε

³ are the state vector, input vector, and process noise, respectively,and f is a nonlinear vector function.

Existing models for the outputs from a grinding process can be convertedinto static functions of the state and input variables. Appendix Aprovides the output equations derived for various outputs such as thegrinding power, roundness, part-size reduction, surface roughness, andwheel diameter. The output equation of the state-space model can bewritten as

y=h(x,u)+ξ(t)  (5)

where y is the output vector, ξ(t) is the measurement noise, and h is anonlinear vector function. According to the two distinct samplingintervals described above, the output vector can be divided into afast-measurement vector, y_(f), and a slow-measurement vector, y_(s)(i.e., y=[y_(f); y_(s)]). The components of y_(f) are real-time sensorsignals of the grinding power and part-size reduction, whereas those ofy_(s) correspond to the roundness, surface roughness, and wheeldiameter, which are measured through post-process inspection.

As in many adaptive filtering schemes, the model parameters are modeledas the random walk processes and then appended to the state vector toform an augmented system:

{dot over (X)}=F(X,u)+η₁(t)  (6)

where X equals [x; θ]; η₁(t) corresponds to [η(t); ν(t)], with η(t) andν(t) being white Gaussian noises; and θ is the vector of modelparameters whose dynamics is given as {dot over (θ)}=ν(t). The outputequation can be represented using the augmented state vector:

y=H(X,u)+ξ(t)  (7)

The augmented system in (6) is represented in the discrete-time domainas follows:

X(i, j+1)=F _(d) [X(i, j)]+η₁(i, j)  (8)

where X(i, j) denotes the state vectors at the jth sampling instance ofthe ith grinding cycle, i (=1, 2, . . . , N) denotes the cycle number,η₁(i, j) is the white Gaussian noise sequence (whose covariance is Q)and F_(d) (X,u)=X+T_(f)F(X,u). Assuming the cycle time of the ith cycle,T_(i), is given by n_(i) (an integer) times the sampling time T_(f)(i.e., T_(i)=n_(i)T_(f)), the sampling index j starts from 0 andincreases up to n_(i)−1 in (8).

A representation of output sampling from a series of grinding cycles isgiven in the discrete time domain as follows:

Within the ith cycle or when jε{0, 1 n_(i)−1}

$\begin{matrix}\begin{matrix}{{y\left( {i,j} \right)} = y_{f}} \\{= {{H_{f}\left\lbrack {{X\left( {i,j} \right)},{u\left( {i,j} \right)}} \right\rbrack} + {\xi_{f}\left( {i,j} \right)}}}\end{matrix} & (9)\end{matrix}$

where H_(f) is composed of the elements in H corresponding to the sensoroutput vector, y_(f), and ξ_(f)(i, j) is the measurement noise (withcovariance R_(f)) in the sensor output.

At the end of the ith cycle or when i=n_(i)

y(i, n _(i))=H[X(i, n _(i)), u(i, n _(i))]+ξ(i,n _(i))  (10)

where ξ(i, n_(i)) is the measurement noise (with covariance R) in thewhole output including the sensor output. Both the slow and fastmeasurements are sampled at the end of the ith cycle.

The observability was tested by linearizing the augmented model in (6)and (7) around more than 10 operating points that were randomly selectedfrom a typical trajectory. Table I summarizes the observability test fortwo estimation tasks, each with two measurement settings. Among theavailable measurements, the grinding power, P and part-size reduction,D_(w), are assumed to be measured with in-process sensors, whereas thewheel diameter, d_(s) and surface roughness, R_(a) would be obtained viapostprocess inspection.

TABLE I OBSERVABILITY UNDER VARIOUS CONDITIONS Variables to Measurementbe estimated setting State Model In-process Postprocess Case variablesparameters sensors inspection Observability 1 x₁, x₂, x₃ — P, D_(w) —Deficient P d_(s) Full 2 x₁, x₂, x₃ R₀ P, D_(w) d_(s) Deficient P d_(s),R_(a) Full

The first task in Table I is to estimate the state variables whileexcluding any model parameters (i.e., X=x). It can be seen from thefirst measurement setting of the task that the system is not observablewhen both P and D_(w) are measured. The estimation becomes feasible whend_(s) is directly measured in addition to P, as shown for the secondsetting. The second case in Table I involves estimating the statevariables in addition to a model parameter in the surface roughnessmodel, R₀; that is, X=[x; R₀]. The output equation in Appendix A forR_(a) is repeated here for reference:

$\begin{matrix}{R_{a} = {\left\lbrack {R_{g} + {\left( {R_{0} - R_{g}} \right){\exp \left( {- \frac{x_{1}}{V_{0}^{\prime}}} \right)}}} \right\rbrack \left( \frac{\pi \; d_{w}x_{2}}{u_{2}} \right)^{\gamma}}} & (11)\end{matrix}$

It is evident from Table I that estimation of R₀ requires a directmeasurement of R_(a). In fact, most parameters in the output equationsrelated to part quality (e.g., surface roughness and roundness) can onlybe made observable through direct feedback, which may not be availableduring a cycle run. The observability analysis in this section providesa strong motivation for involving postprocess measurement data in theestimation of model parameters, as well as full observability of statevariables.

An exemplary estimation scheme is based on extended Kalman filters(EKFs). An EKF operation includes a priori and a posteriori updates ateach sampling instant. The a priori update is made through adiscrete-time simulation of the model, whereas the a posteriori updateinvolves comparing the a priori estimate with the actual measurement. Inthe following descriptions, a vector with a hat (‘̂’) denotes an estimateafter an a posteriori update, whereas one with both a hat and a minussign (‘⁻’) denotes an a priori estimate. Other types of estimationschemes are also contemplated by this disclosure.

During a cycle run, X is estimated using an EKF based on measurement ofy_(f), while y_(s) is estimated by substituting the estimate,{circumflex over (X)}, the known input, u, and a zero noise, ξ=0, intothe output equation. Another EKF operation is applied at the end of eachgrinding cycle based on both the sensor output and the post-processmeasurement, thereby improving the robustness of the overall estimation.The multi-rate EKF operations used in this disclosure are described inmore detail in Appendix B.

At the beginning of a cycle, the actual infeed rate, v (=x₂), startsfrom 0 regardless of its last estimate in the preceding cycle, i.e.{circumflex over (x)}₂ ⁻(i,0)=0. On the other hand, the accumulatedremoval, V_(w)′ (=x₁), by its definition, as well as the wheel diameter,d_(s) (=x₃), should be continuous across cycles. Hence, their estimateshould be also continuous:

{circumflex over (x)} _(1,3) ⁻(i,0)={circumflex over (x)} _(1,3)(i−1,n¹⁻¹)  (12)

where x_(1,3) corresponds to either V_(w)′ or d_(s).

In contrast, model parameter 0 may not be strictly continuous betweenany two cycles in a series due to inherent variations in the grindingprocess. Cycle-to-cycle variations in batch production may be modeled asrandom step changes of the process between cycles. Assuming that thestep variations are purely random, the estimate of the process parameterat the beginning of a cycle is initialized to its last estimate in theprevious cycle as follows:

{circumflex over (θ)}(i,0)={circumflex over (θ)}(i−1, n _(i−1))  (13)

Simulations were performed for the two estimation tasks whoseobservability was tested. The first case involved estimation of statevariables, whereas the second case study involved simultaneous state andparameter estimation for compensating the model-process mismatch.

The simulated process data were generated using (8)-(10) when T_(f)=0.02s from 10 consecutive cycles based on the nominal values of the modelparameters listed in Table II, which were obtained from various studiesthat have involved the grinding of heat-treated steels with aluminumoxide wheels. Although not required by the proposed scheme, an identicalset of grinding conditions was applied to each of the 10 cycles.Specifically, the wheel speed, v_(s), and the work speed, v_(w), werefixed at 37 m/s and 0.533 m/s, respectively, whereas the command infeedrate, u, was scheduled such that plunge grinding is performed in threedistinct stages of roughing, finishing, and spark-out within 17 s(roughing: u=0.0254 mm/s for 0≦t<9.5 s, finishing: u=0.0020 mm/s for9.5≦t<13.3 s, spark-out: u=0 mm/s for 13.3≦t≦17 s). Appropriate processand measurement noises as listed in Table III were added during thesimulation according to (8)-(10).

TABLE II NOMINAL VALUES OF MODEL PARAMETERS IN THE SIMULATION d_(w)d_(s) ₀ K_(s) (mm) (mm) s₀ s₁ (N/mm) δ γ 70 50 49.6 0.08 2380 1 0.2 V₀′R_(g) R₀ (mm³/mm) r_(m) r₀ G₁ G 0.7 3 300 2.4 1 13 0.9

TABLE III PROCESS AND MEASUREMENT NOISES FOR THE SIMULATION Measurementsetting In- Augmented process Postprocess Case state vector, X Processnoise, Q sensors inspection Measurement noise, R A (x₁, x₂, x₃)^(T) 4 ×diag[0.01 10⁻⁹ 10⁻⁹] P, D_(w) — diag[10000 0.0001] P d_(s) diag[1000010⁻⁶] B (x₁, x₂, x₃, R₀)^(T) 4 × diag[0.01 10⁻¹¹ 10⁻⁹ 10⁻⁹] P, D_(w)d_(s) diag[10000 0.0001 10⁻⁶] P d_(s), R_(a) diag[10000 10⁻⁶ 0.0001]

A real-time knowledge of the wheel diameter allows for a tight controlof the work-piece dimension, but in-process sensing of the wheeldiameter is difficult due to the high rotation speed of the grindingwheel and its abrasive action. It is shown above that the wheel diametercannot be estimated based on measurement of either the grinding power,P, or the part-size reduction, D_(w). The main aim in this case studywas to estimate the wheel diameter in real time during a cycle runthrough simulation of the process model based on input variables andestimates of other state variables, while intermittently correcting theestimate based on post-process measurement of its actual value.

FIG. 6 compares the estimation results based on the two measurementsettings with which the observability was tested as Case 1 in Table I.The initial error covariance is denoted as P₀ along with Q denoting theprocess noise covariance for the extended Kalman filter. In FIG. 6,P₀=diag[10 10⁻⁷ 10⁻⁶] and Q=4×diag[0.01 10⁻⁹ 10⁻⁹]. The parenthesesaround d_(s) in the key denote that it is sampled through a post-processmeasurement. In FIG. 6( a) V′_(w)=x₁, in FIG. 6( b) v=x₂, in FIG. 6( c)d_(s)=x₃. FIG. 6( d) is a magnified view of the plot within therectangle in FIG. 6( c).

In the present study, covariance matrices of process noise andmeasurement noise for the Kalman filter were initially determinedaccording to the simulation conditions listed in Table III, and tuned bytrial-and-errors if necessary. In FIG. 6, the solid lines are the truevalues of the state variables, while the other two lines show estimatesof the state variables based on the two measurement settings over aseries of 10 grinding cycles. Note that, for simplicity, FIG. 6 does notshow any idle times between cycles associated with unloading and loadingof parts.

The first measurement setting corresponds to those of existing observersin studies based solely on in-process sensors. FIG. 6 shows thatalthough the first two state variables were tracked well under bothmeasurement settings, the estimated wheel diameter of the firstmeasurement setting exhibits an offset from the true value. In contrast,correcting the estimate of the wheel diameter in the second measurementsetting at the end of each grinding cycle leads to a better overallestimation.

This case was a state-parameter estimation problem with a model-processmismatch in the output equation for the surface roughness. It isdemonstrated that intermittent post-process measurement of the partquality can reduce the model-process mismatch due to process variationsas well as predict the part quality in real time.

In addition to the continuous drift described as the random walkprocess, both batch-to-batch variation and cycle-to-cycle variationswere simulated for parameter R₀ in (11). A batch-to-batch variation wasintroduced by increasing R_(o) by 20% from its value listed in Table IIwhen the first cycle started, whereas a cycle-to-cycle variation wasdescribed as another random walk process by adding a white Gaussiannoise with a covariance of 0.0001 to R₀ at the beginning of every cycle.

The estimation algorithm was applied to the simulated measurement datagenerated according to the above procedure, and input data. Theperformance of the proposed scheme in estimating R_(o) as well as inpredicting the surface roughness at the end of each grinding cycle isshown in FIG. 7. In FIG. 7, the results of estimation are as follows:P_(o)=diag[0.1 10⁻¹¹ 10⁻⁶ 0.01] and Q=1.6×diag[10⁻⁵ 10⁻¹⁴ 10⁻¹² 10⁻⁷].FIG. 7( a) shows model parameter R₀. FIG. 7( b) shows a comparison ofR_(a) and its a priori estimate, {circumflex over (R)}_(a) ⁻, at the endof each cycle.

The true R₀ is shown as a solid line in FIG. 7( a), and the predictionin FIG. 7( b) refers to an a priori estimate of surface roughness at theend of each grinding cycle before an a posteriori update takes placebased on measurement of the actual surface roughness. The measuredsurface roughness in FIG. 7( b) corresponds to that generated bysimulation with a measurement error added according to (7).

Two measurement settings of Case 2 in Table I were considered in thiscase study. As expected from the results of observability test, R₀ inthe output equation for the surface roughness cannot be estimated basedon the first measurement setting. On the other hand, R₀ was updated atthe end of each cycle with the second measurement setting as shown inFIG. 7( a), leading to a good agreement between the measured surfaceroughness and the prediction at the end of each cycle in FIG. 7( b).

The present disclosure has proposed a new control-oriented estimationscheme for a series of grinding cycles in the batch production ofprecision parts. Analysis has revealed that active feedback of thepost-process measurement data allows new and effective observers to bedeveloped, notably in cases where the grinding systems would beunobservable with existing in-process sensors. Although specificapplications have been demonstrated for estimating problems in thegrinding process, this disclosure has focused on introducing thoseinvolved in discrete machining in batches to the new concept ofintegrating all the incoming data flows, with the aim of improvingprocess control. A similar approach could be considered for machiningprocesses in general, as well as polishing and chemical mechanicalplanarization operations for the optics and semiconductor industries.

The systems and methods of the present disclosure may model variationsin the machine tool 102 and/or the parts 106 that arise between discreteprocess cycles. The model that models the variations may include noiseterms that represent the variations. The noise terms may be used toadjust corresponding parameters of the model at the end of a firstoperation. The model may then use the adjusted parameters during asecond operation in order to compensate for the variations that arisebetween the first and second operations. The model that incorporates thenoise term is described hereinafter in further detail.

Multi-rate noise characteristics of discrete process cycles in seriesmay be represented in the state-space format, based on which thepropagation of the error covariance between consecutive cycles isderived. A simulation is carried out to demonstrate the advantage of theproposed change to the estimation algorithm for systems under multi-ratenoise.

A state-space representation in the discrete-time domain may assume thefollowing general structure:

x _(i,k+1) =f(x _(i,k) ,u _(i,k))+η_(i,k)  (14)

where i and k denote indices, x_(i,k)ε

^(n)u_(i,k)ε

^(p) and η_(i,k)ε

^(n) are the state vector, input vector, and within-cycle process noise,respectively, and f is a nonlinear vector function. Note that index idenotes the cycle number while k is the sampling index. The state-spaceequation may be derived from the known physics and prior observation ofthe process. The state variables in Eq. (14), therefore, will correspondto current and past values of physical parameters in the process unlessthey are mapped through state transformations. These physical parametersmay include measurable or immeasurable process variables such as depthof cut, feed, feed rate, and so on in the case of machining processes.Furthermore, model parameters can be appended to the state vector if theprocess is deemed time-varying. The process noise, η_(i,k) is assumed tobe zero-mean white Gaussian with covariance Q_(i,k).

The state variables, thus defined, can be classified into two groupsbased on their characteristics between two cycles. Ignoring anydisturbances between the two cycles, a continuous state variable such asthe machine condition in the (i+1)th cycle would start from their lastvalues of the ith cycle. Furthermore, if the cycle-to-cycle variation ofthe process is ignored, the model parameters appended to the augmentedstate vector will also vary continuously from cycle to cycle. Inreality, any transition of the continuous state variables will bedisturbed by cycle-to-cycle variations such as changes in raw stockproperties and set-up errors. Let x_(i,k) ^(c) denote the vectorincluding all continuous state variables of x_(i,k). Assuming thecycle-to-cycle disturbance is also white Gaussian, the following simplemodel for describing the transition of x_(i,k) ^(c) between twoconsecutive cycles is proposed:

x _(i+1,0) =x _(i,n) _(i) ^(c)+φ_(i)  (15)

where φ_(i) is a noise term. For example, φ_(i) may be a white Gaussiannoise sequence.

In contrast, discontinuous state variables such as the feed rate (in thecase of machining processes) and most of the operating parameters in the(i+1)th cycle will start from their initial conditions, regardless oftheir last values in the ith cycle. Let x_(i,k) ^(d) denote the vectorwhose elements are discontinuous state variables of x_(i,k) wherex_(i,k)=[x_(i,k) ^(c);x_(i,k) ^(d)]. Assuming another independentGaussian noise between two consecutive cycles, the following model isproposed for x_(i,k) ^(d):

x _(i+1,0) =x ₀ ^(d)+ν_(i)  (16)

where x₀ ^(d) is a constant vector representing the initial condition ofthe discontinuous state vector and ν_(i) is a noise term. For example,ν_(i) may be the white Gaussian noise sequence. In this study, Q_(i)denotes the covariance of [φ_(i); ν_(i)]. It can be seen from Eqs.(14-16) that a series of process cycles can be modeled as a system ofdual dynamics, i.e. within-cycle dynamics and cycle-to-cycle dynamicssubject to the multi-rate noise.

The multi-rate estimation algorithm described above was based onextended Kalman filters (EKFs). An EKF operation at each samplinginstance includes a priori and a posteriori updates. The a posterioriupdate refers to correction of state variables and error covariance Pusing the measurement whilst the a priori update is made based on theprocess model. The error covariance P at the beginning of a cyclecontinues from its last value of the previous cycle. This approach,however, falls short of properly addressing the cycle-to-cycle variationthat can be observed at the beginning of each cycle. The propagation oferror covariance between two consecutive cycles considering thecycle-to-cycle noise is derived below.

In the following descriptions, a vector with a hat (‘̂’) denotes anestimate after an a posteriori update, while one with both a hat and aminus sign (‘⁻’) denotes an a priori estimate. A priori update of statebetween two cycles takes place according to Eqs. (15) and (16) asfollows:

$\begin{matrix}{{\hat{x}}_{{i + 1},0}^{-} = \begin{bmatrix}{\hat{x}}_{i,n_{i}}^{c} \\x_{0}^{d}\end{bmatrix}} & (17)\end{matrix}$

As with conventional extended Kalman filters, we assume estimation error{tilde over (x)}=x−{circumflex over (x)} is unbiased. It is desired toobtain:

P _(i−1,0) ⁻ =E└{tilde over (x)} _(i+1,0) ⁻({circumflex over (x)}_(i+1,0) ⁻)^(T)┘  (18)

However, from Eqs. (15), (16) and (17),

$\begin{matrix}{{\overset{\sim}{x}}_{{i + 1},0}^{-} = \begin{bmatrix}{{\overset{\sim}{x}}_{i,n_{i}}^{c} + \phi_{i}} \\\upsilon_{i}\end{bmatrix}} & (19)\end{matrix}$

Here, [φ_(i); ν_(i)] is the white Gaussian noise sequence with Q_(i).Therefore, it can be shown after some manipulation that:

$\begin{matrix}{P_{{i + 1},0}^{-} = {Q_{i} + \begin{bmatrix}p_{i,n_{i}}^{c} & 0 \\0 & 0\end{bmatrix}}} & (20)\end{matrix}$

where P_(i,n) _(i) ^(c)=E└{tilde over (x)}_(i,n) _(i) ^(c)({tilde over(x)}_(i,n) _(i) ^(c))^(T)┘ is a subset of the error covariance at theend of the previous cycle, corresponding to the continuous state vector,x^(c).

With the conventional extended Kalman filter, the error covariance Poften converges to a small value too soon resulting in sluggish responseof estimates to measurements. Considering a series of discrete processcycles is subject to the periodic cycle-to-cycle noise, the prematureconvergence of P, unless prevented by the proposed step in Eq. (20), candegrade the tracking performance of the observer.

FIG. 8 shows the overall schematics of the estimation algorithm. In FIG.8, A denotes the Jacobian matrix of f, C^(f) and C are the Jacobianmatrices of h^(f) and h, respectively, and K^(f) and K are the Kalmangains for y^(f) and y, respectively. Within each cycle, an a prioriupdate of the state vector takes place through a simulation of theprocess model whereas that of error covariance P is carried out afterthe model is linearized around the current state estimate. The aposteriori update is made in two different modes, depending onavailability of the sensor output and postprocess inspection data, bycomparing the a priori estimate of sensor output with the actual sensorsignal. The linearized output equation is used for calculating theKalman gains and the a posteriori update of error covariance P. When anew cycle starts, the a priori updates of state and error covariance Pare made according to Eqs. (17) and (20).

Several parameters including covariances for the measurement noise,R_(i), within-cycle process noise, Q_(i,k), cycle-to-cycle noise, Q_(i),and initial error covariance P₀ may be specified with the proposedobserver. Determining the covariance of measurement noise, R_(i), can bea straightforward task since the measurement accuracy is known in manyapplications. In contrast, the process noise covariances are ratherdifficult to obtain, as they can be time-varying in many processes. Inthis study, both process noise covariances and the initial errorcovariance were determined by trial-and-errors.

Although several systematic methods have been proposed in the literaturefor tuning of process noise covariances of extended Kalman filters(EKF's), achieving such goals by trial-and-error still seems to be acommon practice. However, such ad-hoc methods can be very time-consumingand tedious. Since the proposed observer requires another covariancematrix for the cycle-to-cycle noise (Q_(i)), in addition to thecovariances of conventional EKF's, to be specified, its tuning processcan become even more laborious. Therefore, the following intuitiveguidelines are suggested:

-   -   It is likely that the process will be subject to larger        disturbances and noises when switching from one cycle to the        next than between sampling instances during a cycle run.        Therefore, the cycle-to-cycle noise covariance, Q_(i), should be        larger than the within-cycle process noise covariance, Q_(i,k).        For example, Q_(i) was chosen to be 10,000 times Q_(i,k) for all        three batches of the validation experiment. A similar argument        can be made with respect to the batch-to-batch versus        cycle-to-cycle noises, i.e., a larger covariance matrix should        be chosen for the batch-to-batch process noise. Since the        initial error covariance of the first cycle in each batch, P₀,        can represent the batch-to-batch process noise with any        continuity between consecutive batches ignored, P₀ was chosen to        be 4 times Q_(i) for all three batches of the observer        experiment.    -   Increasing the process noise covariances of the observer led to        quicker responses to measurement updates with increased        sensitivities to measurement noises.

The developed multi-rate estimation scheme was implemented andexperimentally validated for an actual cylindrical grinding process. Thegrinding was performed on a Supertec G20P-45CII cylindrical grindingmachine. Grinding specimens were prepared by heat-treating 4140 steelrods with a nominal work diameter of 63.5 mm to Rockwell hardness C50.In this experimental study, aluminum oxide grinding wheels (32A60 KVBE)with a nominal diameter of 335 mm and a width of 38.1 mm were used. Thewidth of the work-piece was 19.1 mm while the rotational speed of thewheel was fixed at 1800 rpm. A Mitutoyo SJ-201P surface roughness testerwas used to measure the surface roughness over 4 mm in the directionnormal to grinding with a cut-off length of 0.8 mm. The roughness valueof each specimen represents an average of 9 independent measurements.Grinding powers were measured using a fast response PH-3A power cellfrom Load Controls and transferred to a computer through a dataacquisition system at a sampling rate of 200 Hz. The amount of wheelwear was measured by scanning a replica of the wheel profile after eachcycle using a Keyence LK-G10 laser triangulation sensor.

Process models for the cylindrical plunge grinding process weredeveloped based on models and a series of experiments. The processmodels include three dynamic relationships in the grinding process andoutput equations for grinding power, surface roughness, wheel size, andpart size reduction as listed below:

$\begin{matrix}{\overset{.}{v} = {\left( {u - v} \right)/\tau}} & (21) \\{{\overset{.}{V}}_{w}^{\prime} = {\pi \; d_{w}v}} & (22) \\{{\overset{.}{d}}_{s} = {{- \frac{2\; \pi^{g}d_{w}^{1 + g}}{d_{s_{0}}G_{1}}}v_{s}^{- g}v^{1 + g}}} & (23) \\{{P = {{K_{s}\left( {s_{0} + {s_{1}V_{w}^{\prime\delta}}} \right)}d_{w}v}}{R_{a} = {R_{0} + {R_{1}\left( \frac{\pi \; d_{w}v}{v_{s}} \right)}^{\gamma}}}} & (24) \\{D_{w} = {\frac{2V_{w}^{\prime}}{\pi \; d_{w}}\mspace{14mu} {MW}}} & (25)\end{matrix}$

where v is the actual infeed rate (mm/s), u is the command infeed rate,τ is the time constant (s), V_(w)′ is the accumulated amount of metalremoved from the workpiece after wheel dressing or reconditioning(mm³/mm), d_(w) is the nominal diameter of the workpiece (mm), d_(s) isthe wheel diameter (mm), d_(s) _(o) is the initial wheel diameter, v_(s)is the wheel speed (m/s), P is the grinding power (W), R_(a) is thesurface roughness (μm), D_(w) is the accumulated reduction in partdiameter (mm), and G₁, g, s₀, s₁, δ, K_(s), R₀, R₁, and γ are modelparameters.

Nominal values of the model parameters in the above equations weredetermined by curve-fitting the experimental data. The nominal modelparameters, thus obtained, are listed in Table IV. Refer to Appendix Cfor a detailed description of the model development.

TABLE IV Nominal model parameters obtained from experiments K_(s) G₁ Gs₀ s₁ δ N/mm R₀ R₁ γ 87.6 0.0908 1.03 0.00188 0.665 1894 0.478 9.380.776

The process models above were converted into a state-space format forobserver designs. The state vector, thus obtained, includes threevariables, i.e., the accumulated amount of metal removed from theworkpiece after wheel dressing, V_(w)′, the wheel diameter, d_(s), andthe actual infeed rate, v.

The proposed estimation scheme was tested on three batches of grindingcycles. Each of the first two batches consisted of 8 identical grindingcycles in series, emulating a typical batch production run, whereas thelast batch had 10 varying cycles in series. Real-time sensing ofgrinding power and post-process measurement of surface roughness andpart-size reduction were available with all three batches, while theradial wheel wear data were obtained only from the first two batches.

Simultaneous state-parameter estimation problems were formulated byappending model parameters to the original state vector. The modelparameters were assumed to be random walk processes within each cycle.With the first two batches, for example, the continuous state vector ofthe observer is given by x^(c)=(V_(w)′, d_(s), s₀, R₀, G₁)^(T) where s₀,R₀, G₁ are the model parameters, whereas the discontinuous state vector,X^(d), corresponds to the actual infeed rate, v.

In order to demonstrate the advantages of the multi-rate estimation, theperformance of the observer was tested for two different measurementsettings—one with in-process sensing of grinding power P only and theother with both in-process sensing of P and post-process inspection.Note the first measurement setting corresponds to those of existingobservers in studies based solely on in-process sensors for discreteprocesses.

A system is observable if every state can be determined from theobservation of available output variables over a finite time interval. Atest of the observability for the given system shows that the grindingprocess is rendered unobservable when attempting to estimate both themodel parameters and the state variables using only P signals. Feedbackof intermittent post-process measurement of the part quality and toolcondition using the estimation scheme of the present disclosure canovercome such limitations imposed by lack of in-process sensors.

Table V lists observer settings and parameter values used for the firsttwo batches as well as those for the third batch without post-processmeasurement of radial wheel wear. Note the initial values of the modelparameters of the observer were set according to their nominal values inTable IV.

TABLE V Settings and parameters of the observers built for validationBatch 1, 2 3 State vector x^(c) V_(w)′, d_(s), s₀, R₀, G₁ V_(w)′, s₀, R₀x^(d) V v Measurement In-Process P P P P setting Post- — R_(a), d_(s),D_(w) — R_(a), D_(w) process Filter Measurement R_(i,k) ^(f) = 10⁴ R_(i)= diag[10⁴ R_(i,k) ^(f) = 10⁴ R_(i) = diag[10⁴ 10⁻⁴ parameters noise10⁻⁴ 10⁻⁶ 10⁻⁴] 10⁻⁴] Q_(i) diag[100 10⁻⁶ 10⁻⁵ 10⁻³ diag[100 10⁻⁵ 10⁻⁵1000 10⁻⁶] 10⁻⁶] Q_(i,k) 10⁻⁴ × Q_(i) 10⁻⁴ × Q_(i) P₀ 4 × Q_(i) 4 ×Q_(i)

This section presents the performance of the proposed estimation schemeon three batches of grinding cycles. Before each batch starts, thegrinding wheel was dressed according to the dressing parameters used formodel building, i.e., a_(d)=25 μm and s_(d)=0.114 mm.

All grinding cycles in the first two batches were run under identicalgrinding parameters. Specifically, the nominal wheel speed and workspeed were fixed at 31.6 m/s and 0.68 m/s, respectively, while thecommand infeed rate, u, was scheduled such that plunge grinding isperformed in three distinct stages of roughing, finishing, and spark-outwithin 50.6 s (roughing: u=0.0106 mm/s for 0≦t≦27.6 s, finishing:u=0.0021 mm/s for 27.6≦t≦39.6 s, spark-out: u=0 mm/s for 39.6≦t≦50.6 s).

Results of estimation and in-process prediction with the first batch areshown in FIGS. 9-11. Note that idle times between cycles associated withunloading and loading parts and inspection are not shown for simplicity.FIG. 9 presents the results of estimating state variables V_(w)′ and valong with model parameter s₀, which is closely related to the two statevariables according to the process models. FIG. 9( a) shows measuredversus estimated accumulated metal removals. FIG. 9( b) shows estimatedmodel parameter, s_(o). FIG. 9( c) shows command infeed rate u versusestimated actual infeed rates.

In FIG. 9( a), the plus sign represents the measured value of V_(w)′,which can be calculated based on the measured value of D_(w), whereasthe two non-solid lines show the estimates of V_(w)′ based on the twomeasurement settings. When the observer utilized only grinding power Pwhile ignoring the post-process data, the estimated state variableV_(w)′ exhibited an offset from the measured value. Lack of theobservability when relying only on the measurement of P can be explainedby reviewing its model:

P=K _(s)(s ₀ +s ₁ V _(w)′^(δ))d _(w)ν  (26)

Suppose an expectedly high grinding power P is measured due to amismatch between the models and the actual process. The observer willhave to increase the estimate of either V_(w)′ or v to account for thedifference between the predicted P according to the models and themeasured value of P. Since V_(w)′ is an integral of v over time, i.e.,{dot over (V)}_(w)′=πd_(w)ν, any error in the estimated v will result inan increased offset in the estimated V_(w)′. It can be seen thatmeasurement of P is not sufficient for estimating both V_(w)′ and v inthe presence of model-process mismatch or disturbances.

In contrast, correcting the estimate of V_(w)′ at the end of eachgrinding cycle with the proposed multi-rate estimation scheme led to abetter overall estimation. Moreover, since the model parameter, s₀, issimultaneously updated by the multi-rate sampling as shown in FIG. 9(b), the prediction by the process model improves requiring less drasticpost-process corrections as the batch approaches its end in FIG. 9( a).

FIG. 10 shows the performance of the proposed scheme in estimatingparameter R₀ in the output equation for the surface roughness, R_(a), aswell as in predicting R_(a) before each grinding cycle ends. FIG. 10( a)shows estimated model parameter, R₀. FIG. 10( b) shows a comparison ofR_(a) and its a priori estimate, {circumflex over (R)}_(a) ⁻, at the endof each cycle.

It is demonstrated here that intermittent post-process measurement ofthe part quality can reduce the model-process mismatch due to processvariations as well as predict the part quality in real time. The twoestimates of R₀ are shown in FIG. 10( a), and the prediction in FIG. 10(b) refers to an a priori estimate of surface roughness at the end ofeach grinding cycle before an a postriori update takes place based onmeasurement of the actual surface roughness. The parameter R₀ in theoutput equation for the surface roughness cannot be estimated withoutfeedback of the surface roughness. In contrast, R₀ was updated at theend of each cycle with the multi-rate measurement setting as shown inFIG. 10( a), leading to a good agreement between the measured surfaceroughness and the prediction at the end of each cycle in FIG. 10( b).

FIG. 11 compares the results based on the two measurement settings forestimating the wheel diameter. FIG. 11( a) shows measured versusestimated radial wheel wears. FIG. 11( b) shows estimated modelparameter, G₁.

A real-time knowledge of the wheel diameter allows for tight control ofthe work-piece dimension, but in-process sensing of the wheel diameteris difficult due to the high rotation speed of the grinding wheel andits abrasive action. It is demonstrated here that the wheel diameter canbe estimated in real time during a cycle run through simulation of theprocess model based on input variables and estimates of other statevariables, while intermittently correcting the estimate based onpostprocess measurement of its actual value. In FIG. 11( a), theestimated wheel wear based on on-line sensing alone exhibited anincreasing offset from the measured value. In contrast, correcting theestimate of the wheel wear in the multi-rate measurement setting at theend of each grinding cycle led to a better overall estimation, althoughthe estimates were noisy at times due to the high level of measurementnoise. Moreover, updating the model parameter, G₁, improved real-timeestimation of the wheel wear leading to overall decreasing post-processcorrections with increasing cycle numbers.

Results of estimation and in-process prediction with the second batchare shown in FIGS. 12-14. FIG. 12( a) shows measured versus estimatedaccumulated metal removals. FIG. 12( b) shows estimated model parameter,s₀. FIG. 12( c) shows command infeed rate u versus estimated actualinfeed rates. FIG. 13( a) shows estimated model parameter, R_(o). FIG.13( b) shows a comparison of R_(a) and its a priori estimate, h_(a) ⁻,at the end of each cycle. FIG. 14( a) shows measured versus estimatedradial wheel wears. FIG. 14( b) shows estimated model parameter, G₁.

Observations similar to those of the first batch can be made except forthe evident batch-to-batch variations reaffirming the motive forsimultaneous estimation of model parameters. For example, the convergedvalue of model parameter R₀ for the second batch was around 0.27 asshown in FIG. 13( a) whereas that for the first batch was higher at0.33.

The estimation scheme of the present disclosure was applied to anotherbatch consisting of mixed grinding cycles in series. Three differentgrinding schedules, as listed in Table VI, were repeated in tandem,starting with schedule A, until 10 cycles were completed. The wheelspeed and work speed were fixed at 31.5 m/s and 0.65 m/s, respectively.Note the grinding cycle based on schedule C is identical to those of theprevious two batches.

TABLE VI Grinding schedules adopted for the third batch A B C Roughing u= 0.0106 mm/s u = 0.0085 mm/s u = 0.0106 mm/s for 27.6 s for 30 s for27.6 s Finishing u = 0.0021 mm/s u = 0.0042 mm/s u = 0.0021 mm/s for 12s for 15 s for 12 s Spark-out u = 0 mm/s u = 0 mm/s u = 0 mm/s for 5 sfor 11 s for 11 s Cycle time (s) 44.6 56 50.6

Results of estimation and in-process prediction are shown in FIGS. 15and 16. FIG. 15( a) shows measured versus estimated accumulated metalremovals. FIG. 15( b) shows estimated model parameter, s_(o). FIG. 15(c)shows command infeed rate u versus estimated actual infeed rates. FIG.16( a) shows estimated model parameter, R_(o). FIG. 16( b) shows acomparison of R_(a) and its a priori estimate, {circumflex over (R)}_(a)⁻, at the end of each cycle.

FIG. 15 shows that V_(w)′ was tracked well when its estimate wascorrected by post-process data and s₀ was updated simultaneously. Incontrast, the performance in predicting the surface roughness shown inFIG. 16( b) looks inferior to those under fixed input schedules in FIGS.10( b) and 13(b). Evidently, the task of estimation and prediction ismore demanding when the process is subject to varying input schedulesthan when it is run under a series of identical schedules. The conditionof the grinding wheel such as its sharpness, for example, is known tovary over time and converge to a steady-state, which depends strongly oninput grinding parameters. Therefore, increased fluctuations of modelparameters such as R₀ can be expected when the process is run undervarying input schedules. Nevertheless, the prediction performance basedon the multi-rate sampling improved over time, notably after the fifthcycle, in FIG. 16( b).

The proposed algorithm integrates all the incoming data flows, includingsensor signals and post-process measurement data, from a series ofdiscrete process cycles with the aim of improved estimation. In thepresent disclosure, the multi-rate noise characteristics of discreteprocess cycles were represented in a state-space format, based on whichthe multi-rate Kalman filtering algorithm was derived. A new covariancematrix was introduced to naturally represent the cycle-to-cycle noiseand disturbances and a set of intuitive guidelines for tuning of thefilter parameters were issued. Results from implementation of theproposed observer on an actual grinding process demonstrated theapplicability of the proposed multi-rate estimation scheme to practicalproblems in the manufacturing industry. When tested on a series ofidentical grinding cycles, i.e., an emulation of a typical batchproduction run, the implemented multi-rate observer tracked both statesand model parameters well, while a traditional single-rate observerfailed to do so.

The foregoing description of the embodiments has been provided forpurposes of illustration and description. It is not intended to beexhaustive or to limit the invention. Individual elements or features ofa particular embodiment are generally not limited to that particularembodiment, but, where applicable, are interchangeable and can be usedin a selected embodiment, even if not specifically shown or described.The same may also be varied in many ways. Such variations are not to beregarded as a departure from the invention, and all such modificationsare intended to be included within the scope of the invention.

Example embodiments are provided so that this disclosure will bethorough, and will fully convey the scope to those who are skilled inthe art. Numerous specific details are set forth such as examples ofspecific components, devices, and methods, to provide a thoroughunderstanding of embodiments of the present disclosure. It will beapparent to those skilled in the art that specific details need not beemployed, that example embodiments may be embodied in many differentforms and that neither should be construed to limit the scope of thedisclosure. In some example embodiments, well-known processes,well-known device structures, and well-known technologies are notdescribed in detail.

The terminology used herein is for the purpose of describing particularexample embodiments only and is not intended to be limiting. As usedherein, the singular forms “a”, “an” and “the” may be intended toinclude the plural forms as well, unless the context clearly indicatesotherwise. The terms “comprises,” “comprising,” “including,” and“having,” are inclusive and therefore specify the presence of statedfeatures, integers, steps, operations, elements, and/or components, butdo not preclude the presence or addition of one or more other features,integers, steps, operations, elements, components, and/or groupsthereof. The method steps, processes, and operations described hereinare not to be construed as necessarily requiring their performance inthe particular order discussed or illustrated, unless specificallyidentified as an order of performance. It is also to be understood thatadditional or alternative steps may be employed.

APPENDICES Appendix A

Some of the output equations can be obtained by substituting the stateand input variables into analytical models, as follows:

Grinding  power:P = K_(s)(s₀ + s₁x₁^(δ))x₂${{Roundness}\text{:}r} = {{r_{0}\frac{\pi \; d_{w}x_{2}}{v_{w}}} + r_{m}}$${{Surface}\mspace{14mu} {roughness}\text{:}R_{a}} = {\left\lbrack {R_{g} + {\left( {R_{0} - R_{g\;}} \right){\exp \left( {- \frac{x_{1}}{V_{0}^{\prime}}} \right)}}} \right\rbrack \left( \frac{\pi \; d_{w}x_{2}}{u_{2}} \right)^{\gamma}}$

The part-size reduction, D_(w) and wheel diameter, d_(s) directlycorrespond—by their definitions—to two of the state variables, i.e.D_(w)=2x₁/πd_(w) and d_(s)=x₃.

Appendix B

The EKF operates in two distinct modes depending on the two samplingstreams:

Within the ith cycle or when jε{0, 1, . . . , n_(i)−1}

An a priori update takes place according to the discretized model in(8), while the error covariance P is updated according to the followingequation:

P ⁻(i, j)=A(i, j−1)P(i, j−1)A ^(T)(i, j−1)+Q  (B.1)

where A is the Jacobian matrix of F_(d) with respect to X. Once an apriori estimate of the sensor output, ŷ_(f) ⁻, is calculated from (9)with zero noise, the a posteriori update is performed based on itsdifference from the actual sensor output, y_(f), as shown below:

{circumflex over (X)}(i, j)={circumflex over (X)} ⁻(i, j)+K _(f) [y_(f)(i, j)−ŷ _(f) ⁻(i, j)]  (B.2)

P(i, j)=[I−K _(f) C _(f)(i, j)]P ⁻(i, j)  (B.3)

where K_(f)=P⁻(i, j)C_(f) ^(T)(i, j)[C_(f)(i, j)P⁻(i, j)C_(f) ^(T)(i,j)+R_(f)]⁻¹ is the Kalman gain for the fast measurement, C_(f)(i, j) isthe Jacobian matrix of H_(f) with respect to X, and R_(f) is thecovariance of the fast measurement noise, ξ_(f).

At the end of the ith cycle or when j=n_(i)

After the a priori state is estimated, an a priori estimate of the wholeoutput, ŷ⁻ is calculated based on (10) with zero noise. Both the on-linesensor output and the off-line measurement data are used to update thestate estimation as follows:

{circumflex over (X)}(i, n _(i))={circumflex over (X)} ⁻(i, n_(i))+K[y(i, n _(i))−ŷ ⁻(i, n _(i)]  (B.4)

P(i, n _(i))=[I−KC(i, n _(i))]P ⁻(i, n _(i))  (B.5)

where K=P⁻(i, n_(i))C^(T)(i, n_(i))[C(i, n_(i))P⁻(i, n_(i))C^(T)(i,n_(i))+R]⁻¹ is the Kalman Gain and C(i, n_(i)) is the Jacobian matrix ofH with respect to X, and R is the covariance of the measurement noise,ξ.

Appendix C

Process models for the grinding power, surface roughness, part sizereduction, and wheel wear are developed based on a series ofexperiments. A dynamic state-space model for the cylindrical plungegrinding process is derived from these process models.

Appendix C.1

This section describes the procedure for obtaining process models forthe grinding power, surface roughness, and wheel wear from experimentaldata. Consider a cylindrical plunge grinding process, in which arotating cylindrical work-piece with a nominal diameter of d_(w) (m/s)and a surface velocity of v_(w) (m/s) is ground by a rotating grindingwheel with a nominal diameter of d_(s) (mm) and a surface velocity ofv_(s) (m/s). The grinding wheel is fed into the work-piece at a commandinfeed rate, u. Note, due to the mechanical stiffness and sharpness ofthe wheel surface, there exists a dynamic delay of the actual infeedrate, v, in response to the command infeed rate, u, which can bedescribed by a first-order dynamic system:

$\begin{matrix}{{u - v} = {\tau \; \frac{v}{t}}} & \left( {C{.1}} \right)\end{matrix}$

where τ is the time constant (s) whose value is dependent on themachine-wheel-workpiece stiffness, dullness of the wheel, and wheelspeed according to the following equation:

$\begin{matrix}{\tau = \frac{D\; \pi \; d_{w}b_{s}}{{kv}_{s}}} & \left( {C{.2}} \right)\end{matrix}$

where D is the dullness of the wheel, b_(s) is the wheel width (mm), andk is the system stiffness (N/mm). The wheel dullness is known toincrease monotonously with an increase in the accumulated metal removalper unit wheel width after dressing, V_(w)′ (mm³/mm), in general foraluminum oxide wheels as follows:

D=D ₀ +D ₁ V _(w)′_(δ)  (C.3)

where V_(w)′ is the accumulated amount of metal removed from thework-piece after wheel dressing or reconditioning (mm³/mm) while D₀, D₁and δ are constants. A simple model of the grinding power, P (W) for amoderate range of wheel speeds can be represented as a linear functionof the metal removal rate as follows:

P=μDπd_(w)b_(s)v  (C.4)

where μ is the friction coefficient. Combining Eqs. (C.3) and (C.4)yields the grinding power as a function of v and Vw′ as follows:

P=K _(s)(s ₀ +s ₁ V _(w)′^(δ))d _(w)ν  (C.5)

where K_(s)=kμ, s₀=D_(o)πb_(s)/k and s₁=D₁πb_(s)/k .

A straightforward way of determining the model coefficients, K_(s), s₀,s₁ and δ would be to directly fit a set of measurement tuples (P,V_(w)′, v) to Eq. (C.5). However, due to the difficulty in measuring theactual infeed rate, v, an indirect approach was adopted as describedhere. In order to determine K_(s), Eqs. (C.2) and (C.4) were combined asfollows:

P=K _(s)(τv _(s) v)  (C.6)

It can be seen that P is proportional to τv_(s)v with a constant ofproportionality equal to K_(s). Since P is proportional to v when otherparameters are fixed as shown in Eq. (C.5), the response of P to a stepinput u shows characteristics of a first-order system with a timeconstant of τ according to Eq. (C.1). Therefore, τ as well as thesteady-state values of P and v, assuming v equals u at a steady state,were obtained from a step response of P for a known input u. In thismanner, 52 pairs of τv_(s)v and P were obtained from 52 different stepresponses under varying experimental conditions in the following range:

0.0021≦u≦0.0106 (mm/sec)

0≦V_(w)′≦1060 (mm³/mm)

Note that the dressing parameters were fixed at a_(d)=25 um ands_(d)=0.114 mm. FIG. 17 plots the 52 pairs from which K_(s) is given by1894 N/mm. Using the K_(s) thus obtained, FIG. 18 was plotted based onthe 52 step responses after modifying Eq. (C.5) as follows:

$\begin{matrix}{\frac{P}{K_{s}d_{w}v} = {s_{0} + {s_{1}v_{w}^{\prime\delta}}}} & \left( {C{.7}} \right)\end{matrix}$

Through a nonlinear regression analysis of the data in Fig. C.2, s₀, s₁and δ were determined to be 1.03, 0.00188, and 0.665, respectively.

The surface roughness (μm) is known to be dependent on the dressingparameters, the wheel wear and the equivalent chip thickness, h_(eq)(μm), which is defined by the following equation:

$\begin{matrix}{h_{{eq}\;} = \frac{\pi \; d_{w}v}{v_{s}}} & \left( {C{.8}} \right)\end{matrix}$

FIG. 19 plots R_(a) for varying h_(eq) immediately after awheel-dressing operation under fixed dressing parameters. It wasdetermined that the following empirical model would be appropriate fordescribing the relationship between R_(a) and h_(eq) for fixed dressingparameters:

R _(a) =R ₀ +R ₁ h _(eq) ^(γ)  (C.9)

where R₀, R₁ and γ are model parameters whose nominal values aftercurve-fitting are given by 0.478, 9.38, and 0.776, respectively.However, it should be noted the actual surface roughness can be quitedifferent from the prediction of this model, notably when the wheel isworn. Moreover, even the surface roughness after the wheel is dressedunder identical parameters, could vary widely from trial to trial due touncontrolled variations such as those in the condition of the dressingtool. Therefore, there is a strong need for continuously updating thesurface roughness model based on feedback from the process.

The grinding ratio (G-ratio) is defined as the ratio between thevolumetric removal rate of the metal and that of the wheel. In thecylindrical grinding process, the G-ratio, G is given as:

$\begin{matrix}{G = \frac{d_{w}v}{s_{s\; 0}w}} & \left( {C{.10}} \right)\end{matrix}$

where d_(s0) is the initial wheel diameter after dressing (mm) and w isradial wear rate of the wheel (mm/s). The equivalent chip thickness,h_(eq), is a major factor for the G-ratio as shown in the followingmodel:

G=G₁h_(eq) ^(−g)  (C.11)

where G₁ and g are the model parameters. In order to determine thenominal values of the model parameters, the amount of wheel wear wasmeasured after removing metal by 602 mm³ under various h_(eq) values.Since the workpiece width is smaller than that of the wheel, the contactsurface on the wheel is slightly indented after each removal. The depthof indentation was obtained by first making a replica of the wheel usinga steel blade and by scanning the profile of the replica using a lasertriangulation sensor. FIG. 20 shows measured wheel profiles for threedifferent values of h_(eq).

Assuming the G-ratio remains constant over time for a given chipthickness, the G-ratio was obtained from each profile by calculating theratio between the accumulated metal removal in the amount of 602 mm³ andthe volumetric wheel wear as follows:

$\begin{matrix}{G = \frac{V_{w}^{\prime}}{\pi \; d_{s\; 0}\Delta \; r_{s}}} & \left( {C{.12}} \right)\end{matrix}$

where Δr_(s) is the depth of indentation on the wheel. FIG. 21 showsvariation of G-ratio for varying h_(eq), based on which nominal valuesof G₁ and g are given by 87.6 and 0.0908, respectively, viacurve-fitting.

Appendix C.2

This section describes how a dynamic state-space model can be derivedfor the cylindrical plunge grinding process based on its process modelsin Section C.1. Three dynamic relationships can be identified from thedeveloped process models. The first relationship is the dynamic delay ofthe actual infeed rate, v, in response to the command infeed rate, u, inEq. (C.1). The accumulated metal removal, by its definition, is relatedto infeed rate v by another first-order differential equation:

{dot over (V)}_(w)′=πd_(w)v  (C.13)

Moreover, the wheel diameter, d_(s) (mm) as a function of time can berepresented by the following equation by combining Eqs. (C.8), (C.10),(C.11) and {dot over (d)}_(s)=−2w:

$\begin{matrix}{{\overset{.}{d}}_{s} = {{- \frac{2\; \pi^{g}d_{w}^{1 + g}}{d_{s_{0}}G_{1}}}v_{s}^{- g}v^{1 + g}}} & \left( {C{.14}} \right)\end{matrix}$

where d_(s) ₀ is the initial wheel diameter (mm).

Via discretization of Eqs. (C.1), (C.13), and (C.14), a nonlinearstate-space model in the form of x_(k+1)=f(x_(k),u_(k)) can be derivedwhere x=(x₁, x₂, x₃)^(T)=(V_(w)′, d_(s), v)^(T)ε

³ and u=(u₁, u₂)^(T)=(u, v_(s))^(T)ε

². It should be noted that, among the state variables, the actual feedrate, v, is partially discontinuous over a series of machining cyclessince it is reset to 0 at the start of each cycle. On the other hand,the accumulated removal, V_(w)′ by its definition, as well as the wheeldiameter, d_(s), should be continuous across cycles. It should be notedthat both V_(w)′ (tool use) and d_(s) (tool size) represent the toolcondition. Defining a continuous state vector x^(c)=(V_(w)′, d_(s))^(T)and a discontinuous state vector x^(d)=v where x=[x^(c); x^(d)], it canbe seen that a series of grinding cycles is a system with partiallycontinuous states.

It can be seen that many output variables of the grinding process,including those in Section C.1, are nonlinear functions of x and u,i.e., y=h(x,u) when y=(P, R_(a), D_(w), d_(s))^(T) as follows:

$\begin{matrix}{P = {{K_{s}\left( {s_{0} + {s_{1}x_{1}^{\delta}}} \right)}d_{w}x_{3}}} & \left( {C{.15}} \right) \\{R_{a} = {R_{0} + {R_{1}\left( \frac{\pi \; d_{w}x_{3}}{u_{2}} \right)}^{\gamma}}} & \left( {C{.16}} \right) \\{D_{w} = \frac{2x_{1}}{\pi \; d_{w}}} & \left( {C{.17}} \right) \\{d_{s} = x_{2}} & \left( {C{.18}} \right)\end{matrix}$

where D_(w) is the accumulated reduction in part diameter (mm).

1. A method for estimating a process variable associated with a seriesof operations of a manufacturing process, comprising: deriving a modelthat represents a given operation of the manufacturing process, theoperation having first, second, and third process variables associatedtherewith, where the model includes the first, second, and third processvariables, and where variations in the first and second processvariables during each of the operations are substantially immeasurable;measuring the first process variable after a first one of theoperations; measuring the third process variable during a second one ofthe operations using a sensing device; estimating at least one of thefirst and second process variables during the second one of theoperations using the measured first process variable, the measured thirdprocess variable, and the model; and controlling the second operationbased on the at least one of the first and second estimated processvariables.
 2. The method of claim 1, further comprising measuring thethird process variable at predetermined intervals during each of theoperations and measuring the first process variable between each of theoperations.
 3. The method of claim 2, further comprising estimating theat least one of the first and second process variables during each ofthe operations using a state observer.
 4. The method of claim 3, furthercomprising estimating the at least one of the first and second processvariables using a Kalman filter.
 5. The method of claim 1, wherein themodel is further defined as a state space model.
 6. The method of claim5, wherein the first, second, and third process variables arerepresented as functions of state variables of the state space model,and wherein an output vector of the state space model includes themeasured third process variable and the measured first process variable.7. The method of claim 1, wherein the first and second operations areperformed on first and second parts, respectively, using a machine tool.8. The method of claim 7, wherein the model includes a parameter of themachine tool, and wherein the model represents variations in theparameter of the machine tool between the first and second operations.9. The method of claim 8, further comprising: representing thevariations in the parameter using a noise variable of the model;estimating the at least one of the first and second process variablesusing an estimation algorithm that includes gains; modifying one of thegains at the end of the first operation using the noise variable of themodel; and using the modified one of the gains during the secondoperation.
 10. The method of claim 7, wherein the model includes aparameter associated with the first and second parts, and wherein themodel represents variations in the parameter between the first andsecond parts.
 11. The method of claim 10, further comprising:representing the variations in the parameter using a noise variable ofthe model; estimating the at least one of the first and second processvariables using an estimation algorithm that includes gains; modifyingone of the gains at the end of the first operation using the noisevariable of the model; determining the value of the parameter associatedwith the second part based on the value of the parameter associated withthe first part; and using the parameter associated with the second partand the modified one of the gains during the second operation.
 12. Themethod of claim 1, wherein the estimations of the at least one of thefirst and second process variables indicate at least one of ameasurement of a part being produced during the second operation and ameasurement of a tool used to produce the part during the secondoperation.
 13. The method of claim 1, wherein the measured first processvariable includes measurements corresponding to at least one of a partproduced during the first operation and a tool used to produce the partduring the first operation.
 14. A system for estimating a processvariable associated with a series of operations of a machine tool,comprising: an estimation module that includes a model that represents agiven operation of the machine tool, the given operation having first,second, and third process variables associated therewith, where themodel includes the first, second, and third process variables, and wherevariations in the first and second process variables are substantiallyimmeasurable during the given operation; a post-process acquisitionmodule that determines the first process variable after a first one ofthe operations; and an in-process acquisition module that determines thethird process variable during a second one of the operations based onsignals received from a sensing device, wherein the estimation moduleestimates at least one of the first and second process variables duringthe second one of the operations using the determined first processvariable, the determined third process variable, and the model.
 15. Thesystem of claim 14, further comprising an actuation module that actuatesthe machine tool to perform the second one of the operations based onthe at least one of the first and second estimated process variables.16. The system of claim 15, wherein the first process variablerepresents a measurement of at least one of a component of the machinetool and a part produced during the first one of the operations.
 17. Thesystem of claim 16, wherein the machine tool includes a grinding tool,and wherein the first process variable represents at least one of adiameter of a grinding wheel of the machine tool, a residual stressassociated with the part, a roundness of the part, and a surfaceroughness of the part.
 18. The system of claim 15, wherein the signalsreceived from the sensing device indicate an operating condition of themachine tool during the second one of the operations.
 19. The system ofclaim 18, wherein the machine tool includes a grinding tool, and whereinthe signals received from the sensing device indicate at least one of agrinding power of the machine tool and a reduction in the size of thepart.
 20. A method for estimating a process variable associated with aseries of grinding operations of a grinding machine tool during amanufacturing process, comprising: deriving a state space model thatrepresents a given grinding operation of the manufacturing process, thegrinding operation having first, second, and third process variablesassociated therewith, where the state space model includes the first,second, and third process variables, and where variations in the firstand second process variables during each of the grinding operations aresubstantially immeasurable; measuring the first process variable after afirst one of the grinding operations, wherein the measured first processvariable represents a measurement of at least one of a component of thegrinding machine tool and a part produced during the first one of thegrinding operations; measuring the third process variable during asecond one of the grinding operations using a sensing device thatindicates an operating condition of the grinding machine tool during thesecond one of the grinding operations; estimating at least one of thefirst and second process variables during the second one of theoperations using the measured first process variable, the measured thirdprocess variable, and the state space model; and controlling the secondone of the grinding operations based on the at least one of the firstand second estimated process variables.